Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

Abstract

In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.

arXiv comment39 pages, 4 figures. Improved the presentation. Article published in Numerische Mathematik

References in corpus (6)

• Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces
• Approximation error estimates and inverse inequalities for B-splines of maximum smoothness
• Sharp error estimates for spline approximation: explicit constants, $n$-widths, and eigenfunction convergence
• Robust multigrid solvers for the biharmonic problem in isogeometric analysis
• Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations
• Optimal spline spaces for $L^2$ $n$-width problems with boundary conditions

Cited by in corpus (12)

• Mathematical foundations of adaptive isogeometric analysis
• A comparison of smooth basis constructions for isogeometric analysis
• An Interior Penalty coupling strategy for Isogeometric non-conformal Kirchhoff-Love shell patches
• Construction of $C^2$ cubic splines on arbitrary triangulations
• Goal-Adaptive Meshing of Isogeometric Kirchhoff-Love Shells
• Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties, graded meshes & variational crimes
• Best low-rank approximations and Kolmogorov n-widths
• A theoretical study on the effect of mass lumping on the discrete frequencies in immersogeometric analysis
• Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations
• Macro-element Refinement schemes for THB-Splines: Applications to Bézier Projection and Structure-Preserving Discretizations
• Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions
• A characterization of linear independence of THB-splines in ${\mathbb{R}}^n$ and application to Bézier projection